What is the smallest prime factor of 5^8+10^6–50^3?

5^8 has 5 as a prime factor.

10^6 has 2 and 5 as prime factors.

50^3 has 2 and 5 as prime factors.

Thus only 2 and 5 are possible answers. The answer is 2 if the solution is even and 5 if the solution is odd. Eliminate BDE.

5^8 is odd because products of odd integers are always odd.

10^6 and 50^3 are even because products of even integers are always even.

Odd + Even - Even = Odd.

The answer is C.

Hi HaseebR7,

You have the right idea, but you should double-check your math.

Hi CCMBA,

You have to very careful when trying to apply "math theory" to complicated situations. If your understanding is off (even a little bit), then you'll get one of the wrong answers and not even know it.

This question is really about re-writing exponents and finding ways to combine 'like' terms. Try using exponent rules against this question...

GMAT assassins aren't born, they're made,
Rich

Okay, so I tried to recombine and this is what I got.

5^8 - 10^6 + 50^3

5^8 - (5^6 * 2^6) + (5^3 * 5^3 *2^3)

5^8 - 5^6(2^3 * 2^3) + (5^6 * 2^3)

5^8 - [5^6 * 2^3](2^3 + 1)

5^6 * 5^2 - {5^6 * 2^3](2^3 + 1)

5^6 [5^2 - (2^3 *2^3 + 1)]

5^6 (25 - 65)

5^6(-40)

-5^6 * 2^2 * 2 *5 --> answer is 2?

I kept getting turned around, but I *think* the math is finally right. I would be guessing on this question. I think recognizing that the answer should be 2 or 5 is enough.

EDIT: Nope, I just cheated by breaking out the calculator and got a multiple of 3. Chetan2u makes it look so easy!

There is an error in your concept:

Smallest prime factor of 2*5*11 is 2 - Correct
Smallest prime factor of 2 + 5 is neither 2 nor 5 but actually 7 - a new prime number.
When you add, you don't know the prime factors you will get.

Here is how you will recombine:

\(5^8 + 10^6 - 50^3\) (You have the negative sign misplaced in your expression)

\(5^8 + 2^6*5^6 - 2^3*5^6\)

\(5^6 * (25 + 64 - 8)\)

\(81 * 5^6\)

\(3^4 * 5^6\)

The smallest prime is 3.

(5^8)+(10^6)–(50^3)
Breaking each component down gives:
(5^8) smallest prime is 5
(2*5)^6 smallest prime is 2
(2*5^2)^3 smallest prime is 2

Answer: A is that correct?

okay never mind, I need to decompose the equation further so that it turns into (5^6)(81). Hence, 3...

Smallest prime factor will be 3.

Simplify the equation in the same way as posted by many.

option B it is.

VERITAS PREP OFFICIAL SOLUTION:

With exponents questions like this, your guiding principles should include "Find Common Bases" and "Factor" - the composite bases 5, 10, and 50 can all be turned into prime bases of 5s and 2s, and the addition and subtraction can be factored to multiplication. First you should break the 10 and 50 down into prime factors:

5^8+2^6*5^6−2^3*5^6
Then factor the common 5 terms out to create a multiplication problem:

5^6(5^2+2^6−2^3)
Which allows you to deal with the math in the parentheses, since those numbers are all reasonable to calculate by hand:

5^6(2^5+6^4−8)
Which leads to 5^6(81), which factors down to (5^6)(^34). Since those are all prime, and the question asks for the SMALLEST prime, the answer is 3.

Answer: B.

\(5^8 + 10^6 – 50^3\)

\(= 5^8 + 5^6 * 2^6 – 5^6 * 2^3\)

\(= 5^6(25+64) - 5^6 * 2^3\)

\(= 5^6(25+64-8)\)

\(= 5^6 * 81\)

\(= 5^6 * 3^4\)

Smallest prime factor = 3

So did I....missed minus.!

Shouldn't the part in red above be -2^3 and not +2^3?

hi pacifist,
you are correct...
it was a typing error... edited it thank you

5^8 + 10^6 - 50^3
5^8 + (5x2)^6 - (5^2 x 2)^3
5^8 + (5^6)(2^6) - (5^6)(2^3)
5^6(5^2 + 2^6 - 2^3)
5^6(81)

81 --> 3 is lowest prime

B.

\(5^8+10^6–50^3\)

= \(5^8+ ( 5^6*2^6 ) – ( 5^6*2^3)\)

= \(5^6 \ [ \ 5^2+ 2^6 – 2^3 \ ]\)

= \(5^6 \ [ \ 5^2+ 2^3 ( 2^3 – 1 ) \ ]\)

= \(5^6 \ [ \ 25+ 8 * 7 \ ]\)

= \(5^6 *81\)

= \(5^6 *3^3\)

Thus , the smallest prime factor will be (B) 3

\(5^8 + 10^6 - 50^3\)

\(5^8 = (5^2)^4\)

\(10^6 = ((2*5)^2)^3\)

\(50^3 = (2*5^2)^3\)

\((5^2)^3 + (5^2)^3 - (5^2)^3 [5^2 + 2^6 - 2^3] = 5^6 * 81 = 5^6 * 3^4.\)

So, \(3\) is the smallest prime factor. Ans - B.

the expression can be rewritten as:
\(5^8+(5^6 * 2^6)-(5^3 * 10^3) = 5^8 + (5^6 * 2^6)-(5^3*5^3*2^3) = 5^8 + (5^6 * 2^6)- (5^6 * 2^3)\)
which reduces to:
\(5^6(5^2+2^6-2^3)= 5^6(25+64-8)=5^6(81)=5^6 * 3^4\)
thus 3 is the smallest prime factor

The key here is :to reduce the expression in order to find its basic prime components only then can we decide which is the smallest prime factor
ANSWER: B

You can do it by Remainders also-

checking by 2 =
Remainder of 5^8 by 2 = 1
Remainder of 10^6 by 2 = 0
Remainder of 50^3 by 2 = 0

Thus By Remainder theorum : 1+0-0 = 1 NOT divisible by 2

Checking by 3-
Remainder of 5^8 by 3 = 1
Remainder of 10^6 by 3 = 1
Remainder of 50^3 by 3 = 2

Thus By Remainder theorum : 1+1-2 = 0 , hence the same is divisible by 3 , THUS CHOICE B